On the Iitaka volumes of log canonical surfaces and threefolds
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910521335218176 |
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| author | Chen, Guodu Han, Jingjun Liu, Wenfei |
| author_facet | Chen, Guodu Han, Jingjun Liu, Wenfei |
| contents | Given positive integers $d\geqκ$, and a subset $Γ\subset [0,1]$, let $\mathrm{Ivol}_{\mathrm{lc}}^Γ(d,κ)$ denote the set of Iitaka volumes of $d$-dimensional projective log canonical pairs $(X, B)$ such that the Iitaka--Kodaira dimension $κ(K_X+B)=κ$ and the coefficients of $B$ come from $Γ$. In this paper, we show that, if $Γ$ satisfies the descending chain condition, then so does $\mathrm{Ivol}_\mathrm{lc}^Γ(d,κ)$ for $d\leq 3$. In case $d\leq 3$ and $κ=1$, $Γ$ and $\mathrm{Ivol}_\mathrm{lc}^Γ(d,κ)$ are shown to share more topological properties, such as closedness in $\mathbb{R}$ and local finiteness of accumulation complexity. In higher dimensions, we show that the set of Iitaka volumes for $d$-dimensional klt pairs with Iitaka dimension $\geq d-2$ satisfies the DCC, partially confirming a conjecture of Zhan Li.
We give a more detailed description of the sets of Iitaka volumes for the following classes of projective log canonical surfaces: (1) smooth properly elliptic surfaces, (2) projective log canonical surfaces with coefficients from $\{0\}$ or $\{0,1\}$. In particular, the minima as well as the minimal accumulation points are found in these cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07391 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Iitaka volumes of log canonical surfaces and threefolds Chen, Guodu Han, Jingjun Liu, Wenfei Algebraic Geometry 14J27, 14B05, 14E30 Given positive integers $d\geqκ$, and a subset $Γ\subset [0,1]$, let $\mathrm{Ivol}_{\mathrm{lc}}^Γ(d,κ)$ denote the set of Iitaka volumes of $d$-dimensional projective log canonical pairs $(X, B)$ such that the Iitaka--Kodaira dimension $κ(K_X+B)=κ$ and the coefficients of $B$ come from $Γ$. In this paper, we show that, if $Γ$ satisfies the descending chain condition, then so does $\mathrm{Ivol}_\mathrm{lc}^Γ(d,κ)$ for $d\leq 3$. In case $d\leq 3$ and $κ=1$, $Γ$ and $\mathrm{Ivol}_\mathrm{lc}^Γ(d,κ)$ are shown to share more topological properties, such as closedness in $\mathbb{R}$ and local finiteness of accumulation complexity. In higher dimensions, we show that the set of Iitaka volumes for $d$-dimensional klt pairs with Iitaka dimension $\geq d-2$ satisfies the DCC, partially confirming a conjecture of Zhan Li. We give a more detailed description of the sets of Iitaka volumes for the following classes of projective log canonical surfaces: (1) smooth properly elliptic surfaces, (2) projective log canonical surfaces with coefficients from $\{0\}$ or $\{0,1\}$. In particular, the minima as well as the minimal accumulation points are found in these cases. |
| title | On the Iitaka volumes of log canonical surfaces and threefolds |
| topic | Algebraic Geometry 14J27, 14B05, 14E30 |
| url | https://arxiv.org/abs/2407.07391 |